Uniquely identifiable state-space and ARMA parametrizations for multivariable linear systems

Multivariable systems can be represented, in a uniquely identifiable way, either by canonical forms or by so-called overlapping forms. The advantage of the latter is that they do not require the a priori estimation of a set of structural invariants (e.g. Kronecker invariants). We show here how to define uniquely identifiable overlapping parametrizations for state-space and ARMA models. We show that these parametrizations are all related to a set of intrinsic invariants, which are obtained from the Markov parameters of the system. Different forms of overlapping ARMA parametrizations are derived and their properties discussed.     

On the robustness of multivariable linear feedback systems in state-space representation

A singular values-based approach to specify the robustness of a multivariable linear feedback system in state-space representation is investigated. The robustness measure which is considered is the largest spectral norm of an additive uncertainty in the closed-loop system matrix, for which stability is guaranteed. It is shown that under the constraint of prescribed pole placement, a lower bound for the robustness measure is maximized, when the Frobenius norm of the closed-loop system matrix is minimized.     

Canonical state space representations for multivariable systems

The problem of state space representations for multiple-input, multiple-output linear time-invariant systems is considered. The known definitions of the first and second canonical forma for scalar systems are used to facilitate the derivation of explicit canonical expressions for multivariable systems.     

The stability of linear multivariable systems

A general criterion is presented for establishing the stability of a multivariable system from the stability of the simplified system consisting of its diagonal elements. This new criterion includes all known criteria of stability based on the diagonal system elements such as Rosenbrock's (1974), Nwokah's (1980) anil Koussiouris's (1980). For the class of matrices satisfying this criterion an equivalent result in terms of eigenloci can also be stated     

The design of linear multivariable systems

This paper describes a computer-aided procedure whereby a succession of single-loop designs, using Nyquist loci, yields a multivariable design which is stable and attenuates disturbances. The procedure permits more freedom than previously available for the selection of the compensating matrix. It is shown how a simple modification of the procedure enables security against component failure to be obtained.     

Model reduction for linear multivariable systems

A method has been proposed for obtaining reduced-order models for multivariable systems. Termed nonminimal partial realization, this method is shown to encompass the methods of aggregation (or eigenvalue preservation) and moments matching (Padé-type approximation about more than one point). In particular, a promising subclass of reduced models, called aggregated partial realizations, is discussed in detail and appears to combine the good points of the aggregation as well as the moments matching methods.     

Canonical forms for linear multivariable systems

In this paper a class of well-known canonical forms for single-input or single-output controllable and observable systems are extended to multivariable systems. It is shown that, unlike the single-variable case, the canonical forms are generally not unique, but that the structure of the canonical form can be controlled to some extent by the designer. A major result of the paper is that a multi-input system can be transformed to a set of coupled single-input subsystems.     

Compensator design for multivariable linear systems

The problem of the specification of the order and structure of a linear dynamic compensator in order to obtain arbitrary pole placement in a closed-loop linear system comprised of the compensator in cascade with a linear plant is discussed. A significant application of the theory is to the design of optimal systems in those cases where not all the state variables of the plant can be measured. These results permit a completely algorithmized approach to the design of compensators for linear systems.     

Compensator design for linear multivariable systems

A configuration and an orderly procedure for the design of linear multivariable control systems are presented. This procedure employs transfer function matrices to specify the system response to set point changes and disturbances. Neither a state-variable formulation nor solution of the matrix Riccati equation are required. The design procedure is illustrated by a simple example involving an unstable system.     

On multivariable linear systems

A general result in linear system theory concerning the controllability and observability of multi-input multi-output systems is presented. Specifically, it is shown that any controllable observable linear system can be made controllable from any input and observable from any output using output feedback.     

Near-decoupling of linear multivariable systems

Available exact-decoupling conditions by constant compensation do not address the important class of ‘nearly decouplable’ systems. For practical systems, parameter perturbations, measurement errors and system-modelling inaccuracies may conceal the system decouplability by constant- or low-order compensation, thus calling for unnecessarily high-order dynamics. In this paper, state-space decoupling conditions formulated as rank-degeneracy conditions are used to detect how close a system is to one that is exactly decouplable. The procedure simultaneously provides the compensators necessary to achieve the indicated level of decoupling. The conditions are extended to cases where it is required to use a fixed constant compensator for decoupling under different sets of perturbations     

The systematic design of control systems for large multivariable linear time-invariant systems

A systematic design procedure for determining realistic feedforward-feedback control systems for large multivariable linear constant systems is outlined in this paper. A numerical example of a boiler system and a distillation column is included and a comparison of the simulated response of the controlled systems obtained by the proposed method is made with the corresponding conventional control systems. The new control systems obtained are relatively simple, are realistic to implement, and appear to be a significant improvement over the conventional ones.     

Geometric state-space theory in linear multivariable control: A status report

With the renewed emphasis in control theory on qualitative structural issues, as distinct from techniques of optimization, the last decade has brought significant growth in the application of geometric ideas to the formulation and solution of problems of controller synthesis. In this article we review the status of geometric state space theory as developed for application to systems that are linear, multivariable and time-invariant. After a brief summary of the underlying geometric concepts ((A, B)-invariant subspaces and (A, B)-controllability subspaces) we outline two standard problems of feedback control that have been successfully attacked from this point of view, and briefly survey recent results in a range of other topic areas. Then we discuss certain issues of methodology, and conclude with some remarks on computational procedures.     

Dyadic feedback laws for linear multivariable systems

The extent to which it is possible to assign the eigenvalues of the system Where K is constrained to have the dyadic structure. K = qp^T, is investigated. It is shown that a necessary and sufficient condition for the existence of a dyadic feedback law to assign arbitrary eigenvalues, in so for as this can be accomplished by any state variable feedback law, is that the system matrix of the controllable part of the system is non-derogatory. The proof furnishes a constructive procedure for computing a vector q in this case. The treatment is based on the Anderson-Luenberger approach, and a now result is presented regarding the information which can be extracted from their canonical form.     

Observers for linear multivariable systems with applications

This paper presents an algorithm for the design of asymptotic state estimators (observers) for index-invariant uniformly observable time-varying linear finite-dimensional multivariable systems. The results obtained indicate that asymptotic estimators can be employed in optimally designed regulators provided an increase from the optimal cost is tolerable. It is also shown that any uniformly observable and uniformly controllable plant with index-invariant observability and controllability matrices can be stabilized with an observer.     

Robust controllers for uncertain linear multivariable systems

This paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems. In the first part, it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties, and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty. In the second part, in contrast with the first part, it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known. The type of plant uncertainty is the so-called “stable-factor” uncertainty, and necessary and sufficient conditions are given for robust stabilization. In the third part, the model of uncertainty is a ball in the space of rational matrices metrized by the so-called graph metric, and sufficient conditions for robust stabilization are derived.     

线性多变量系统的控制Lyapunov函数构造

提出线性多变量系统控制Lyapunov函数(CLF)构造的一般方法. 先证明可以通过解一类Lyapunov方程, 得到线性系统二次型的CLF. 接着证明了对于线性系统, 这种方法可以提供所有二次型的CLF. 最后证明了若线性系统存在CLF, 那么必存在二次型的CLF. 由此完全解决了线性系统的CLF构造问题.     

The feedforward control of linear multivariable time-invariant systems

Necessary and sufficient conditions are derived for a stable feedforward control system to exist for a multivariable linear time-invariant system so that: (1) any measurable disturbances occurring in the system are asymptotically rejected, and (2) the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs. The result may be interpretated as being a generalization of the single-input, single-output feedforward servo problem to multivariable systems or as being a solution to the asymptotic decoupling problem. Some numerical examples varying from 2nd to 9th order are included to illustrate the result.